Related papers: A central limit theorem for scaled eigenvectors of…

Let $d$ be a fixed large integer. For any $n$ larger than $d$, let $A_n$ be the adjacency matrix of the random directed $d$-regular graph on $n$ vertices, with the uniform distribution. We show that $A_n$ has rank at least $n-1$ with…

We consider nonintersecting random walks satisfying the condition that the increments have a finite moment generating function. We prove that in a certain limiting regime where the number of walks and the number of time steps grow to…

Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency…

We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of "homogenization" for Dyson Brownian Motion, this yields the universality of quantities which depend on…

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both…

We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the…

We study random k-lifts of large, but otherwise arbitrary graphs G. We prove that, with high probability, all eigenvalues of the adjacency matrix of the lift that are not eigenvalues of G are of the order (D ln (kn))^{1/2}, where D is the…

The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of $n$ mutually independent and identically distributed random variables with finite first and…

We study the adjacency matrices of random $d$-regular graphs with large but fixed degree $d$. In the bulk of the spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$ down to the optimal spectral scale, we prove that the Green's…

The paper proves several limit theorems for linear eigenvalue statistics of overlapping Wigner and sample covariance matrices. It is shown that the covariance of the limiting multivariate Gaussian distribution is diagonalized by choosing…

We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets…

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd{\H o}s-R{\'e}nyi graph $G(N,p)$. Recently, it was shown by Lee, up to an…

In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\rightarrow \infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm…

We observe that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. Namely, we derive recurrent relations for the limiting averaged moments of the adjacency…

A latent space model for a family of random graphs assigns real-valued vectors to nodes of the graph such that edge probabilities are determined by latent positions. Latent space models provide a natural statistical framework for graph…

We study how eigenvectors of random regular graphs behave when projected onto fixed directions. For a random $d$-regular graph with $N$ vertices, where the degree $d$ grows slowly with $N$, we prove that these projections follow…

The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs $\G_N, N\ge 1$ to some target graph $W$.…

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We…

We present a new notion of limits of weighted directed graphs of growing size based on convergence of their random quotients. These limits are specified in terms of random exchangeable measures on the unit square. We call our limits…

We prove a moderate deviation principle for subgraph count statistics of Erdos-Renyi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an…